First, we use Lemma 8 with \(F=\widetilde{F}_n\)  and \(G=\Phi ^{\widetilde{P}}_{m,n}\) , \(T=\widetilde{L}_{m+1,n}^{-1}\)  and Lemma 7 with \(G=\widetilde{F}_n-\Phi ^{\widetilde{P}}_{m,n}\)  from [1]}, each applied with conditional Fourier transforms and finite \(m\) -th conditional moment. Next, \({\frac{\mathrm {d}^{}}{\mathrm {d}x^{}}}\Phi ^{\widetilde{P}}_{m,n}=\phi ^{\widetilde{p}}_{m,n}\) , thus by eq.phitp,eq.tp-bound,
\(\big (1+|x|\big )^m \Big | {\frac{\mathrm {d}^{}}{\mathrm {d}x^{}}}\Phi ^{\widetilde{P}}_{m,n}(x)\Big |= \big (1+|x|\big )^m \big | \phi ^{\widetilde{P}}_{m,n}(x)\big |= \big (1+|x|\big )^m \Big | \phi (x)+\sum _{r=1}^{m-2}\widetilde{p}_{r,n}(x) \Big |\le c(m)\) 
